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MIT 3 052 - X-RAY INVESTIGATIONS OF CHAIN-MOLECULES IN SOLUTION

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X-RAY INVESTIGATIONS OF CHAIN-MOLECULES IN SOLUTION BY O. KRATKY and G. POROD (Institute for Theoretical and Physical Chemistry, Graz University)21. Qualitative treatment of the scattering of chain molecules. A dumbbell-shaped molecule is to be hit by an X-ray that propagates normally to its axis. As everybody knows, this gives rise to an interference-like interaction of the scattered waves of the two atoms. If the distance between the atoms equals D then the first intensity maximum lies in the scattering direction 2 ϑ where sin 2ϑ = λ/D (1) If there are many such pairs present in irregular distances then the scattered intensities are simply adding up. Even if the distance is not exactly D = λ/sin 2 ϑ we still find a noticeable interference effect in the direction 2ϑ. Very generally, we will be able to say that at arbitrary constitution of the agglomerates particles present in a size range of D will give rise to a typical interference effect in the direction 2 ϑ which is coupled to D through (1). We will name the region of D that has been assigned to the angle 2 ϑ by relation (1) the critical range of this angle. Every scattering angle 2 ϑ describes distances of a certain order of magnitude in a typical manner. Generally, every particle will interfere with any other particle in the object under consideration; but those particles present in a critical range interact differently than through bare addition of intensities, but rather in a typical manner corresponding to their mutual position. To get to know the diffraction in a certain direction 2 ϑ one has to imagine the critical range around every single atom and to let interfere all the atoms situated in it. It is clear, of course, that the critical range is not a sharply defined space: but we will see at least that such a treatment has advantages for the discussion of the scattering of a chain molecule. Now, when we turn to this problem we recognize that there are five typical size ranges that have to be distinguished: I. The single atoms. II. The regularly self-repeating basic building blocs. III. Regions of the chains that still can be considered as straight. IV. Larger regions of the chain already containing the variety of all orientations, i.e. which are coiled. V. The whole chain-like molecule.3The region I will not appear purely when using the common wavelengths in the order of magnitude of single atoms because the critical range D penetrates already into the order of magnitude of the interatomic distances at the largest angles. Then the closer packed atoms corresponding to the monomer will appear at medium angles. Here the regular repetition of the main distance of the monomers will cause a step in the intensity curve which will be the more prominent the straighter the monomers are ordered, i.e. the weaker the coiling is. Hence, the position of the step does not depend on the total size of all ordered monomers combined but on the size of the single monomers or the main distance of two neighboring [monomers]. The higher or lower regularity in the repetition of this distance does not change the position of the interference effect but its sharpness, its special course. An interference effect at smaller angles can be assigned to the ranges of the chain as a whole that are still straight (size range III) and which can be described at its first approximation as the scattering effect of a straight stick. As will be shown below, the angle dependence of this range can be described with 1/ϑ. The scattering at still smaller angles has to be assigned to the size range IV, i.e. the already coiled parts of the molecules. It is approximately described by a course corresponding to 1/ϑ2. - The size range regarding the whole molecule (size range V) comprises a statistical accumulation whose diffraction is given by a Gaussian curve (Guinier1)). 1. The new statistics of the coiling. Before we go into detail about the already mentioned size ranges with an exact calculation we have to make some remarks concerning the shape of chain-molecules. The description of chain-molecules developed by Haller2), Guth and Mark3), Kuhn4) and others has been extended namely by the last researcher. To describe the grade of coiling he used the term preferential length Am. It is our task to replace the real chain with a model in such a way that we add N straight elements of the length Am in any arbitrary orientation (random trajectory) where the following conditions have to be fulfilled: 1 ) A. Guinier. Theses Serie A. Nr. 1854 (1939). Nr. d’Ordre 2721. 2) W. Haller. Kolloid-Z. 56, 257 (1932). 3) e.g. W. Guth and H. Mark. Monatsh. Chem. 65, 93 (1934). 4) e.g. W. Kuhn and H. Kuhn. Helv. Chim. Acta 26, 1394 (1943).41. The total length AmN of the replacement model shall be equal to the so called hydrodynamic length L of our chain-molecule which is its length in the completely uncoiled state. 2. The square root of the square mean distance between start point and end point shall be equal for the chain-molecule and for the model. If the chain-molecule already contains a diversity of all orientations we will always be able to chose the length Am of the straight pieces in an unequivocal way so that both conditions are met, that model and chain-molecule correspond each other with regard to their total length and the distance of the end points in the coiled state. The stronger the coiling is the smaller is the preferential length Am as can be seen immediately and clearly from Figure 1. Fig.1: Coiled chain-molecule and model which is created by adding of lines with a preferential length Am in random orientations. In the sense of this treatment we can say, of cause, that two points of the chain that are separated by a piece of the hydrodynamic length L have a coiling defined mean distance Am(L/Am)½: the number of preferential lengths necessary to replace this part of the chain is L/Am; the square root of this number, i.e. (L/Am)½, has to be multiplied by the preferential length Am to obtain the linear mean distance of the two points in the sense of random orientation statistics. If we ask for the “contraction” of this chain piece one has to form the quotient Q of the linear mean distance of the5two points (LAm) ½ and the hydrodynamic length of the segment between the two points L. We immediately obtain: Q = (Am/L)½ (2) Because L = NAm where N means the number of preferential lengths contained in L then Q = N-½ At hydrodynamic


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MIT 3 052 - X-RAY INVESTIGATIONS OF CHAIN-MOLECULES IN SOLUTION

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